Think of the word boost as meaning "help up." Maybe you need a boost getting over a fence (hmm — are you sneaking in or sneaking out?), or maybe you are feeling low and need to boost your spirits — chocolate, anyone?
Sometimes words appear that really don't have any clear origins. Boost is one of those words, but it's a fun word, and it's thought that maybe it has some connection to boose, a Scots dialect word that carries the idea of a push. A slang meaning of the word is to sneakily steal, like shoplifting. You gave your cousin a boost into the window, not knowing he was planning on boosting some sneakers.